a: \(\frac{4xy-5}{10x^3y}-\frac{6y^2-5}{10x^3y}\)
\(=\frac{4xy-5-6y^2+5}{10x^3y}=\frac{4xy-6y^2}{10x^3y}\)
\(=\frac{2y\left(2x-3y\right)}{10x^3y}=\frac{2x-3y}{5x^3}\)
b: \(\frac{7x+6}{2x\left(x+7\right)}-\frac{3x+6}{2x^2+14x}\)
\(=\frac{7x+6}{2x\left(x+7\right)}-\frac{3x+6}{2x\left(x+7\right)}\)
\(=\frac{7x+6-3x-6}{2x\left(x+7\right)}=\frac{4x}{2x\left(x+7\right)}=\frac{2}{x+7}\)
c: \(\frac{1}{x-5x^2}-\frac{25x-15}{25x^2-1}\)
\(=\frac{-1}{x\left(5x-1\right)}-\frac{25x-15}{\left(5x-1\right)\left(5x+1\right)}\)
\(=\frac{-5x-1-x\left(25x-15\right)}{x\left(5x-1\right)\left(5x+1\right)}=\frac{-5x-1-25x^2+15x}{x\left(5x-1\right)\left(5x+1\right)}\)
\(=\frac{-25x^2+10x-1}{x\left(5x-1\right)\left(5x+1\right)}=\frac{-\left(5x-1\right)^2}{x\left(5x-1\right)\left(5x+1\right)}=\frac{-\left(5x-1\right)}{x\left(5x+1\right)}\)
d: \(\frac{x+1}{x-3}-\frac{1-x}{x+3}-\frac{2x\left(1-x\right)}{9-x^2}\)
\(=\frac{x+1}{x-3}+\frac{x-1}{x+3}-\frac{2x\left(x-1\right)}{\left(x-3\right)\left(x+3\right)}\)
\(=\frac{\left(x+1\right)\left(x+3\right))+\left(x-1\right)\left(x-3\right)-2x\left(x-1\right)}{\left(x-3\right)\left(x+3\right)}\)
\(=\frac{x^2+4x+3+x^2-4x+3-2x^2+2x}{\left(x-3\right)\left(x+3\right)}=\frac{2x+6}{\left(x-3\right)\left(x+3\right)}\)
\(=\frac{2\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}=\frac{2}{x-3}\)
